Course 424

Group Representations Ia

Dr Timothy Murphy

Seminar Room Friday, 24 January 2003 15:00–17:00

Attempt 6 questions. (If you attempt more, only the best 6 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are finite, and all representations are of finite degree over ℂ.

1. Define a group representation. What is meant by saying that the representation α is simple?

Show that every simple representation of G is of degree ≤ |G|.

Determine all simple representations of the quaternion group Q_{8} = {±1, ±i, ±j, ±k} (up to equivalence) from first principles.

2. What is meant by saying that the representation α is semisimple?

Prove that every representation α of a finite group G (of finite degree over ℂ) is semisimple.

Define the intertwining number I(α, β) of 2 representations α, β.

Show that the simple parts of a semisimple representation are unique up to order.

3. Define the character χ_{α}(g) of a representation α.

Explain how an action of a group G on a finite set X gives rise to a (permutation) representation α of G.

Show that

χ_{α}(g) = |{x ∈ X : gx = x}|.

Determine the characters of S_{4} defined by its actions on the set X = {a, b, c, d} and the set Y consisting of the 6 subsets of X containing 2 elements.

Hence or otherwise draw up the character table of S_{4}.

4. Show that if the simple representations of G are σ_{1}, …, σ_{s} then

dim^{2} σ_{1} + ⋯ + dim^{2} σ_{s} = |G|.

Determine the degrees of the simple representations of S_{5}.

5. Show that a simple representation of an abelian group is necessarily of degree 1.

Prove conversely that if every simple representation of G is of degree 1 then G must be abelian.

Show that the simple representations of an abelian group G themselves form a group (under multiplication) isomorphic to G.

6. Draw up the character table of D_{5} (the symmetry group of a regular pentagon).

Determine also the representation ring of D_{5}, ie express each product of simple representations of D_{5} as a sum of simple representations.

7. Draw up the character table of the alternating group A_{4}.

8. By considering the eigenvalues of 5-cycles, or otherwise, show that S_{n} has no simple representations of degree 2 or 3 if n ≥ 5.